Conformally Osserman manifolds

TL;DR

This paper classifies conformally Osserman manifolds, showing that in most dimensions they are locally conformally equivalent to Euclidean space or rank-one symmetric spaces, based on their Weyl tensor properties.

Contribution

It provides a classification of conformally Osserman manifolds in dimensions other than 3, 4, and 16, linking their structure to well-known geometric spaces.

Findings

Conformally Osserman manifolds are classified in most dimensions.

Such manifolds are locally conformally equivalent to Euclidean or rank-one symmetric spaces.

The classification excludes dimensions 3, 4, and 16 due to special geometric properties.

Abstract

An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension $n \ne 3, 4, 16$ is locally conformally equivalent either to a Euclidean space or to a rank-one symmetric space.